We review an scenario for the non-equilibrium dynamics of glassy systems thathas been motivated by the exact solution of simple models. This approach allowsone to set on firmer grounds well-known phenomenological theories. The oldideas of entropy crisis, fictive temperatures, free-volume... have cleardefinitions within these models. Aging effects in the glass phase are alsocaptured. One of the salient features of the analytic solution, the breakdownof the fluctuation-dissipation relations, provides a definition of a bonafide{\it effective temperature} that is measurable by a thermometer, controls heatflows, partial equilibrations, and the reaction to the external injection ofheat. The effective temperature is an extremely robust concept that appears innon-equilibrium systems in the limit of small entropy production as, forinstance, sheared fluids, glasses at low temperatures when quantum fluctuationsare relevant, tapped or vibrated granular matter, etc. The emerging scenario isone of partial equilibrations, in which glassy systems arrange their internaldegrees of freedom so that the slow ones select their own effectivetemperatures. It has been proven to be consistent within any perturbativeresummation scheme (mode coupling, etc) and it can be challenged byexperimental and numerical tests, some of which it has already passed.